Extensive Simulations for Longest Common Subsequences Finite Size Scaling, a Cavity Solution, and Connguration Space Properties

Given two strings X and Y of N and M characters respectively, the Longest Common Sub-sequence LCS Problem asks for the longest sequence of non-contiguous matches between X and Y. Let LN be the length of a LCS of two random strings of size N. Using extensive Monte Carlo simulations for this problem, we nd a nite size scaling law of the form ELN=N = S + AS=ln N p N + :::, where S and AS are constants depending on S, the alphabet size. We provide precise estimates of S for 2 S 15. We also study the related Bernoulli Matching model where the diierent e n tries of the strings" are matched independently with probability 1 =S. Let L B N M be the length of a longest sequence of matches in this case, for a given instance of size N M. On the basis of a cavity-like analysis we nd B S r = 2 p rS, r , 1=S , 1, where B S r is the limit of EL B N M =N as N ! 1 , the ratio r = M= N being xed. This formula agrees very well with our numerical computations of EL B N M. It provides also a v ery good approximation for Sr, the corresponding function of the random string model, the approximation getting better as S increases. We nally study the ground state" properties of this problem. We nd that the numberNLCS of solutions typically grows exponentially with N. In other words, this system does not satisfy Nernst's principle". This is also reeected at the level of the overlap between two LCSs chosen at random, which is found to be self averaging and to aproach a deenite value qS 1 a s N ! 1 .